Noname manuscript No. (will be inserted by the editor) Guided Dive for the Spatial Branch-and-Bound D. Gerard · M. K¨ oppe · Q. Louveaux September 12, 2016 Abstract We study the spatial Brand-and-Bound algorithm for the global opti- mization of nonlinear problems. In particular we are interested in a method to find quickly good feasible solutions. Most spatial Branch-and-Bound-based solvers use a non-global solver at a few nodes to try to find better incumbents. We show that it is possible to improve the branching rules and the node priority by exploiting the solutions from the non-global solver. We also propose several smart adaptive strategies to choose when to run the non-global solver. We show that despite the time spent in solving more NLP problems in the nodes, the new strategies enable the algorithm to find the first good incumbents faster and to prove the global opti- mality faster. Numerous easy, medium size as well as hard NLP instances from the Coconut library are benchmarked. All experiments are run using the open source solver Couenne. Keywords spatial branch-and-bound, local search, heuristic, guided dive, branching, Couenne 1 Introduction A wide range of engineering and scientific applications may be written as a non- linear optimization problem, e.g. electric [29], water [24] and gas [35] distribution networks, contigency analysis in electric networks [13], nuclear reactor reloading D. Gerard Department of Electrical Engineering and Computer Science, University of Li` ege, Li` ege 4000, Belgium E-mail: [email protected]M. K¨ oppe Department of Mathematics, University of California, Davis, CA 95616, U.S.A. E-mail: [email protected]Q. Louveaux Department of Electrical Engineering and Computer Science, University of Li` ege, Li` ege 4000, Belgium E-mail: [email protected]
Transcript
Noname manuscript No.
(will be inserted by the editor)
Guided Dive for the Spatial Branch-and-Bound
D. Gerard · M. K
¨
oppe · Q. Louveaux
September 12, 2016
Abstract We study the spatial Brand-and-Bound algorithm for the global opti-mization of nonlinear problems. In particular we are interested in a method to findquickly good feasible solutions. Most spatial Branch-and-Bound-based solvers usea non-global solver at a few nodes to try to find better incumbents. We show thatit is possible to improve the branching rules and the node priority by exploitingthe solutions from the non-global solver. We also propose several smart adaptivestrategies to choose when to run the non-global solver. We show that despite thetime spent in solving more NLP problems in the nodes, the new strategies enablethe algorithm to find the first good incumbents faster and to prove the global opti-mality faster. Numerous easy, medium size as well as hard NLP instances from theCoconut library are benchmarked. All experiments are run using the open sourcesolver Couenne.
Keywords spatial branch-and-bound, local search, heuristic, guided dive,branching, Couenne
1 Introduction
A wide range of engineering and scientific applications may be written as a non-linear optimization problem, e.g. electric [29], water [24] and gas [35] distributionnetworks, contigency analysis in electric networks [13], nuclear reactor reloading
D. GerardDepartment of Electrical Engineering and Computer Science, University of Liege, Liege 4000,BelgiumE-mail: [email protected]
M. KoppeDepartment of Mathematics, University of California, Davis, CA 95616, U.S.A.E-mail: [email protected]
Q. LouveauxDepartment of Electrical Engineering and Computer Science, University of Liege, Liege 4000,BelgiumE-mail: [email protected]
2 D. Gerard et al.
patterns [36], reinforced concrete beams optimization [10], lens design in optics[22]. Many applications can also be found in chemical engineering [6], such aschemical processes control [7], life cycle optimization for sustainable design [20]and production planning in multi-plant facilities [23]. Other applications includechannel interference [11] and optimal resource management [39] in wireless net-works, optimal path for vehicles [1,21], tra�c modeling [18], conflict resolutioninvolving multiple aircraft [37] and flight clearance [15].
Definition 1 Smooth nonlinear programming (NLP) problems are convenientlyexpressed as
min f(x)
s.t. c(x) 0
Li
xi
Ui
i 2 N
(NLP(L,U ))
where N = {1, . . . , n}, f : Rn ! R and c : Rn ! Rm are twice continuouslydi↵erentiable functions.
Any (NLP(L,U )) problem can be solved by a NLP solver. The NLP solversare categorized depending on the methods implemented: sequential quadratic pro-gramming [8], interior point [32], penalty [14], augmented lagrangian [34], trust-region [9], filter [17]. . . Those strategies guarantee the global convergence to a localoptimum, i.e. the convergence to a point satisfying the KKT conditions from anystarting point.
If (NLP(L,U )) is nonconvex, the local optimum towards which a NLP solverconverges may not be the global optimum. The local optimum objective value mayalso be far from the global optimum. Making sure to find the global optimum of aproblem is computationally demanding.
The best-known method for solving nonconvex NLP problems to global opti-mality is the spatial Branch-and-Bound (sBB) [38,40], implemented for instance inBARON [41], Couenne [3], ANTIGONE [27], LindoGlobal [26], SCIP [5], ↵-BB [2]and COCONUT [33]. The sBB explores the whole feasible set of the problem. Inall sBB implementations, the solvers make use of the best feasible solution knownso far to prune some regions which cannot contain any better solution. Hence,finding good feasible solutions quickly is of paramount importance to prune asmuch space as possible. This paper focuses on new variants of the sBB algorithmto find good feasible solutions quicker. All the ideas are implemented in Couenne,a well-known open-source global optimization software.
The urge behind obtaining good feasible solutions quickly finds its root inMixed-Integer Programming (MIP). Some MIP models remain very hard to solveto optimality. In such cases, several heuristics are known to sometimes providegood feasible solutions quickly. Variable Neighborhood Search (VNS) [28] seeks afeasible solution in a neighborhood and subsequently adds a perturbation to moveto another neighborhood. Local Branching [16] explores the neighborhood of afeasible solution by increasing or decreasing the value of a few integer variableswhile excluding the previously found incumbent with a so called ’no-good cut’. Re-laxation Induced Neighborhood Search (RINS) [12] is based on the intuition thatthere is a link between the incumbent and the solution of the continuous relax-ation. Many variables take the same values between the two. Therefore, RINS fixes
Guided Dive for the Spatial Branch-and-Bound 3
them and solves a sub-MIP on the remaining variables to hopefully find a solutionwhich achieves both integrality and a good objective value. Relaxation EnforcedNeighborhood Search (RENS) [4] explores the neighborhood of the solution of thecontinuous relaxation by rounding all integer variables. [12] also introduces theguided dive, which uses the current incumbent solution to decide to dive in theleft or right child node first.
The heuristics in MIP gave birth to several heuristics for nonconvex MixedInteger Nonlinear Programming (MINLP). [25] uses VNS for the global searchphase and a black-box convex MINLP solver for the local search. [31] argues thata local search performed with a convex MINLP solver is significantly slower thanit was in MIP. Therefore, [31] proposes to avoid the convex MINLP solver andconsider a sequence of limited-size MILPs and NLPs instead. The NLP estimatesthe descent direction of the original objective function and the MILP solves aconvexification with a local branching constraint. [30] also performs a sequence ofNLPs and MIPs, but with a rounding of the integer variables. However, researchstill lacks insights on how to adapt these heuristics for nonconvex NLP withoutany integer variables, besides using a black-box NLP solver.
We base our work on the aforementioned works, in particular the guided diveof [12], and aim at extending it to nonconvex NLP.
The sBB recursively partitions the feasible set into subsets such that everysolution to (NLP(L,U )) is feasible in one of the subsets.
Each subset is associated with a node uniquely defined by a specific set ofbounds (l, u) such that L
i
li
ui
Ui
for each i 2 N , and corresponds to theNLP
min f(x)
s.t. c(x) 0
li
xi
ui
i 2 N
(nodeNLP(l,u))
The feasible set of (nodeNLP(l,u)) is
⌦(l, u) = {x 2 Rn : c(x) 0; li
xi
ui
; i = 1, . . . , n} (1)
Definition 2 At a node (nodeNLP(l,u)), consider a variable xb
, b 2 N on which
we branch, and let {x(1)b
, . . . , x(K+1)
b
} be a sequence of branching values such that
x(1)
b
= lb
, x(K+1)
b
= ub
and x(k)
b
x(k+1)
b
for k = 1, . . . ,K. The multiway branching
defines the K partitions of ⌦(l, u), also referred to as child nodes, as
⌦(k) = ⌦(l, u) \ {x 2 Rn : x(k)b
xb
x(k+1)
b
} k = 1, . . . ,K (2)
⌦(k) is the feasible set of nodeNLPk
(l, u).
In practice, all the newly created nodes are stored in a queue Q.
Definition 3 Let Q be a set of nodes {nodeNLPj
(l, u)} to solve for j = 1, . . . , |Q|,among which a sBB implementation must choose the next node to proceed. A node
selection strategy assigns a priority p(j) to each node in Q. The next node to selectis nodeNLP
j
(l, u) with j 2 argmaxj
p(j).
4 D. Gerard et al.
Most sBB implementations also factorize the original problem (nodeNLP(l,u))into a series of equivalent constraints for which a convex relaxation can be derived[19].
One iteration of the sBB algorithm involves selecting a problem (nodeNLP(l,u))2 Q with the highest priority p (Definition 3). Its convexification is solved, whichyields the point x(l,u). If x(l,u) 62 ⌦(l, u), the node is partitioned according toDefinition 2. (nodeNLP(l,u)) may also be solved with an NLP solver to obtain alocal optimum h(l,u).
Upon partitioning a set into two subsets or more, one may use for instance thebest bound node comparison strategy to decide which branch is to be explored first.However, so far, there is no NLP-specific rule to guess which branch to explore firstto find feasible solutions with a good objective value quickly. This paper aims atintroducing new such rules and provides extensive tests to assess how e↵ective theyare on real-world and academic problems. In particular, Couenne runs IPOPT (anNLP solver) on (nodeNLP(l,u)) at a few upper levels of the sBB tree to try to findbetter incumbents. We show that it is possible to improve the branching rules byexploiting the solution of (nodeNLP(l,u)) provided by the NLP solver.
The remainder of this paper is organized as follows. Section 2 introduces theguided dive for MIP and proposes an extension for nonconvex NLP. Section 3incorporates the guided dive and its specificities into a multiway branching scheme.Finally, Section 4 provides extensive tests and conclusions for all the featuresintroduced. Section 5 concludes and gives directions for further research.
2 Guided dive
First introduced by [12] for MIP, the guided dive consists in a small modification ofthe tree traversal strategy. In the branch-and-bound branching procedure, the firstdecision is the selection of a variable to branch on, and the second is the choice ofwhich child node to explore first. Let x⇤ be the best incumbent known so far, andxb
be the selected branching variable at node (nodeNLP(l,u)). The guided dive
explores first the child of (nodeNLP(l,u)) such that x⇤b
2 [x()b
, x(+1)
b
], accordingto Definition 2. The guided dive and the depth-first tree traversal, by makingthe Branch-and-Bound dive towards the best incumbent and then backtrack thetree, are together somewhat similar to local branching heuristics which explicitlyexplore the neighborhood of x⇤.
It is important to point out that although x⇤b
2 [x()b
, x(+1)
b
], the best in-cumbent x⇤ may not be feasible in (nodeNLP(l,u)), or in any of its child nodesnodeNLP
k
(l, u), k = 1, . . . ,K. Unlike local branching heuristics, the guided divealso makes use of the incumbent even if it is not located in the neighborhood ofx⇤, provided x⇤
b
2 [x()b
, x(+1)
b
] only. The incumbent could therefore be far from⌦(l, u). [12] notes that despite its simplicity, this approach is quite e↵ective atimproving the solutions found in MIP. We extend the guided dive to the sBB fornonconvex NLP problems.
In Section 4, we first show computationally that as for MIP problems, theguided dive leads to a major improvement for nonconvex NLP problems solvedto global optimality. The first limitation of the original guided dive is that there
Guided Dive for the Spatial Branch-and-Bound 5
are many nodes which cannot be guided if x⇤b
62 [lb
, ub
]. A blind child selectionmust therefore be performed. In practice, the solvers use a random or a fixed childselection. Couenne belongs to the latter by diving in the leftmost child first.
In the sBB, an NLP solver is run at several nodes at the upper levels of the tree,each potentially providing one NLP feasible solution. Finding feasible solutionsmay be hard. In this paper, we consider problems for which at least a few feasiblesolutions are found throughout the sBB. Let F (l,u) be the set of all known feasiblepoints when node (nodeNLP(l,u)) is processed. We improve the guided dive byusing the points in F (l,u), which includes
1. former incumbents before a new better incumbent was found;2. newly computed feasible points which are not better than the incumbent.
At node (nodeNLP(l,u)), with the branching variable xb
, let H(l,u) = {x 2F (l,u) : l
b
xb
ub
} be the set of known feasible points for which the value ofxb
lies within the bounds [lb
, ub
] of (nodeNLP(l,u)). Our improved guided divechooses the feasible point h(l,u) 2 H(l,u) with the best objective value to direct thedive at node (nodeNLP(l,u)).
In some cases, the feasible point h(l,u) used by one node (nodeNLP(l,u)) can bereused in some of its child nodes, thereby avoiding to solve some new NLP prob-lems. Denote by the child node of (nodeNLP(l,u)) such that h(l,u)
b
2 [x()b
, x(+1)
b
],
according to Definition 2. There are two cases to consider. Case 1 considers h(l,u) 2⌦(l, u) whereas case 2 considers h(l,u) 62 ⌦(l, u).
Case 1. h(l,u) 2 ⌦(l, u) and h(l,u)
b
2 [x()b
, x(+1)
b
] imply that h(l,u) 2 ⌦(), the
feasible set of nodeNLP
(l, u). h(l,u) can be reused by nodeNLP
(l, u), for anybranching variable selected when processing nodeNLP
(l, u). For the other childnodes, h(l,u) cannot be reused if the branching variable is x
b
and sometimes cannotbe reused for the other branching variables x
g
, g 2 N\{b} if h(l,u)
g
is excluded fromthe tightened bounds [l
g
, ug
] by the bound tightening procedures.
Case 2. h(l,u) 62 ⌦(l, u) and h(l,u)
b
2 [x()b
, x(+1)
b
] imply that h(l,u) can only bereused by nodeNLP
(l, u) provided its branching variable is also xb
. For the otherbranching variables of nodeNLP
(l, u) as well as for all the branching variables ofthe other child nodes, there is no garantee that h(l,u) can be reused.
In most sBB implementations, at the v upper levels of the tree, the NLPproblem is always solved by the NLP solver, starting from the solution of theconvexification. The aim is to find new incumbents early. Below the v level, i.e. forall the lower levels of the tree, no NLP problem is solved.
At the beginning of the tree, the major source of feasible points comes fromrunning a local NLP solver at several nodes, the other one being solutions of theconvexification provided they are nonlinear feasible. However, solving local NLPproblems has a non-negligible running time. We further improve the algorithmwith the main goal of solving a NLP problem when it is most useful.
In our implementation, we replace the v threshold with the v1
and v2
thresholdsinstead (v
1
v2
). Above the v1
level, the NLP problem is always solved becausethe partitioning and the bounds tightening techniques at the upper levels of thetree are likely to enable the NLP solver to find new incumbents. Between the v
1
6 D. Gerard et al.
and v2
levels, the NLP problem is solved if H(l,u) = ;. It is more useful to solvea NLP problem if H(l,u) = ;, because the solution obtained will direct the diveinstead of having to perform a blind diving. Below the v
2
level, no NLP problemis solved. However, if H(l,u) 6= ; for a node (nodeNLP(l,u)) below the v
2
level,then one feasible point h(l,u) 2 H(l,u) is used for the child selection; otherwise thedefault blind diving is performed.
3 Multiway branching
Couenne implements the branching procedure of Definition 2 forK = 2. By default,for continuous variables, the branching value is x
b
= ↵x(l,u)
b
+ (1 � ↵)xMb
, where
x(l,u) is the solution of the convex relaxation of (nodeNLP(l,u)) and xMb
= lb+ub2
.This branching-point selection is refered to as mid-point. The branching is per-formed by adding the inequality x
b
xb
for the left child nodeNLP1
(l, u) andxb
� xb
for the right child nodeNLP2
(l, u). There are several other branching-pointselection rules described in [3]. All of them aim at improving the convexificationof the subsequent child nodes. However, none of them adapt their branching valueto try to dive as fast as possible towards regions with higher odds of finding betterfeasible solutions. The previous section argued that diving in the child node con-taining the incumbent has a positive impact. This section explores modifications ofthe branching values to try to achieve even better performance. For the remainderof this section, we rewrite h(l,u) as h.
Figure 1 and Figure 2 both illustrate a disadvantageous 2-way branching. Inboth figures, the left child is the vertically hatched region on the left and the rightchild is the horizontally hatched region on the right. In Figure 1, the branchingvalue is x
b
. With the mid-point selection xb
= ↵x(l,u)
b
+(1�↵)xMb
and the default
value ↵ = 0.25, the relative distance |xMb �¯
xb|ub�lb
is at most ↵
2
= 0.125, placing xb
rela-
tively close to the center xMb
= lb+ub2
. However, the solution hb
of (nodeNLP(l,u))may lie anywhere in the range [l
b
, ub
], regularly resulting in situations like the onedepicted in Figure 1, where h
b
is close to ub
. According to its definition, the guideddive chooses the right child first because it contains h
b
. However, the span [xb
, ub
]is fairly wide, which prevents a fast dive towards the close neighborhood of h.
Figure 2 illustrates another partitioning where another branching value delim-its a relatively short span containing h
b
. Diving in the right child first enablesthe guided dive to explore the neighborhood of h quickly. However, dismissing x
b
as the branching point has the major drawback of not focusing on improving theconvexification of the child nodes anymore, which was the purpose of x
b
. Hence theconvexification of the vertically hatched area is barely improved. While the parti-tioning of Figure 2 may help find new incumbents faster, it impedes on the globalconvergence of the sBB. We propose to take these considerations into account bykeeping the branching value x
b
and further partitioning with a second branchingvalue, as defined in Definition 4 and depicted in Figure 3 (we can assume x
b
hb
with no loss of generality).
Definition 4 The guided 3-way partitioning defines a set of 2 branching values{x
b
, hb
��1
} (resp. {xb
, hb
+�2
}) which delimit the 3 parts according to Defini-tion 2. The guided 3-way partitioning creates the second branching point eitheron the left (h
b
��1
) or on the right (hb
+�2
) of hb
, depending on the location of
Guided Dive for the Spatial Branch-and-Bound 7
lb xb hb ub
1 2
Fig. 1: Branching value xb
lb xb hb ub
1 2
Fig. 2: Branching value close to hb
xb
and hb
. �1
and �2
are margins which are a fraction of ub
� lb
. The branchingvalue h
b
��1
is possible if xb
hb
��1
and the branching value hb
+�2
is possibleif h
b
+�2
ub
. In Definition 3, the 3 parts are each given a priority p such thatp(3) > p(2) > p(1) (resp. p(2) > p(3) > p(1) with {x
b
, hb
+�2
}), i.e. the 3rd childis to be solved before the 2nd child, which is to be solved before the 1st child.
lb xb hb ub
1 2 3�1
Fig. 3: Guided 3-way partitioning with branching values {xb
, hb
��1
}
Definition 5 extends the guided 3-way by allowing one additional branchingpoint on the other side of h
b
. Illustration in Figure 4.
Definition 5 The guided 4-way partitioning defines a set of 3 branching values{x
b
, hb
��1
, hb
+�2
} which delimit the 4 parts according to Definition 2. �1
and�
2
are margins which are a fraction of ub
� lb
. In Definition 3, they are each givena priority p such that p(3) > p(2) > p(4) > p(1)
lb xb hb ub
1 2
3
4�1�2
Fig. 4: Guided 4-way partitioning
8 D. Gerard et al.
Finally, we define the guided 2-way partitioning in Definition 6.
Definition 6 The guided 2-way partitioning defines {xb
} as the sole branchingvalue which delimits the 2 partitions according to Definition 2. In Definition 3,they are each given a priority p such that p(1) > p(2) if h
b
2 [lb
, xb
], and p(2) > p(1)otherwise.
Creating more nodes in a sBB results in a noticeable computing overhead,more than in MIP. For each sBB node, several bound tightening algorithms areperformed, a convexification is computed and solved, and an NLP solver is possiblyrun. This suggests that the 3-way and a fortiori the 4-way partition should bescarcely used. At each node (nodeNLP(l,u)), the 2-way, the 3-way or the 4-way ischosen wisely to avoid creating too many small nodes. From the 4-way partitioning(Definition 5), if h
b
��1
�xb
< � for some value � � 0 (i.e. part 2 is too small), parts2 and 3 are merged, thereby resulting in a 3-way partitioning. If u
b
� hb
��2
< �
(i.e. part 4 is too small), parts 3 and 4 are merged. The priority p(k) is set suchthat p(3) > p(2) (if not merged) > p(4) (if not merged) > p(1).
4 Computational results
Our features are implemented in Couenne 0.4 and benchmarked on an Intel Core i7at 3.47GHz. Several di↵erent versions of the diving strategies were tested on NLPinstances from the Coconut library. Among the whole Coconut test set (around600 problems), some problems were discarded:
– 258 problems solved almost instantaneously;– 12 problems claimed by IPOPT to be infeasible because it makes Couenne
assume that the problem is infeasible and stop;– 26 problems for which numerical issues were encountered by the original Couenne
or our modified versions;– 78 problems with no solution found in the sBB within the two hours time limit,
besides the local solution found by IPOPT.
This leaves us with 170 usable problems for testing, further split into
– 117 easy or medium size problems;– 53 hard problems.
A problem is considered hard if at least the original Couenne or one of our modifiedversions does not converge to global optimality within the two hours time limit.Moreover, besides the local solution found by IPOPT, at least one feasible solutionwith a better objective value must be found in the sBB by the original Couenneor one of our modified versions. If no better solution can be found in the sBB, thebranching strategies could not be compared.
Our measurements are:
1. total running time (resp. total number of processed nodes) until proof of globaloptimality;
2. time (resp. number of processed nodes) to close 60% of the gap between thefirst feasible solution f
0
found at root node and the global optimum f⇤.
Guided Dive for the Spatial Branch-and-Bound 9
We compare the original Couenne (C) with our modified versions (M). Thetime gain between the two algorithms is tM�tC
max(tM ,tC)
, where t is either the total
running time or the time to close 60% of the gap. If the time gain is negative, themodified version is faster than Couenne. Otherwise Couenne runs faster than ourmodified verison. To compare the total number of nodes or the number of nodesprocessed to close 60% of the gap, the formula nM�nC
max(nM ,nC)
is used, where n is thenumber of nodes.
4.1 Guided dive
Table 1 shows the average gains over the easy and medium size problems for thedi↵erent diving strategies. Test 1 (first row in Table 1) shows the gain if Couennedives first in the child node nodeNLP
(l, u) containing the best incumbent x⇤, i.e.x⇤ 2 ⌦(l, u), which is similar to a local branching heuristic. If x⇤ 62 ⌦(l, u), thedefault diving of Couenne is performed. Finding feasible solutions helps closing the60% gap quicker. However, there is little impact on the total running time, besidesa few branches closed thanks to the incumbent. Generating tighter convexificationshas a much larger impact on the total running time. We refer the reader to [3].
Test 2 is the guided dive as originally defined in [12] for MIP, which dives in the
child node nodeNLP
(l, u) such that x⇤b
2 [x()b
, x(+1)
b
], even if x⇤ 62 ⌦(). As forMIP problems, the guided dive also leads to a significant gain for NLP problems,with respect to diving in the child node containing x⇤. To show how importantthe guided dive is, the inverted guided dive in test 3 first dives in the child nodenodeNLP
⌧
(l, u) such that x⇤b
62 [x(⌧)b
, x(⌧+1)
b
], the opposite of test 2. The slightlypositive gain values for the inverted guided dive mean that it is slightly worse thanthe default diving of Couenne, which dives in the leftmost child first.
In test 4, the feasible point considered for the diving is either computed at thecurrent node or inherited from a parent node, not necessarily the best incumbentknown so far. The gain is not as good as the guided dive in test 2. This points outthat it is more advantageous to use the best incumbent instead of several otherfeasible points with a weaker objective value, although the latter allow the sBB touse the guided dive more often.
In test 5, we use the best incumbent if possible as in test 2; otherwise we useanother feasible point inherited from a parent node as in test 4. If x⇤
b
62 [lb
, ub
], thebest incumbent cannot be used and a blind dive must be performed. To avoid thisas much as possible, we use another known feasible solution from a parent or thesolution from a local NLP solver at the current node, if solved. This improves theguided dive based only on the best incumbent.
In test 6, from H(l,u) = {x 2 F (l,u) : lb
xb
ub
}, the set of known feasiblepoints for which the value of x
b
lies within the bounds [lb
, ub
], we consider thefeasible point h(l,u) 2 H(l,u) with the best objective value, no matter its locationin the variable space. This further slightly improves the performance. Since h(l,u)
may be spatially far from ⌦(l, u), it is reasonable to try promote a solution h(l,u) 2F (l,u) closer to ⌦(l, u), albeit with a weaker objective value. A weighting betweenthe value of the objective function and the distance has been carried out andbenchmarked. However, all the weighting values tested yield a deterioration of thegain. It seems that the feasible solution with the best objective value should alwaysbe used first, no matter how far it is from ⌦(l, u).
10 D. Gerard et al.
Test 7 uses the two depth thresholds v1
and v2
. The values used are v1
= d0.57eeand v
2
= d0.91ee, where e is the number of variables of the extended reformulation[19]. Between the v
1
and v2
levels, the NLP problem is not solved if H(l,u) 6= ;.The sBB tree is expected to barely be a↵ected by not computing those solutions.Indeed, the total number of nodes and the number of nodes to close 60% of thegap are almost unchanged. However, computing fewer local NLP problems savestime. The time gain depicted in Table 1 is rather limited because we also used aparameter preventing Couenne from solving more than 10 local NLP problems inthe whole tree, so the algorithm of test 7 only avoids a few calls to IPOPT. Thisparameter, among others, was used to improve the original Couenne. If one wereto increase this parameter, the improvement from test 6 to test 7 would be larger,but the gains of all the other tests, from tests 1 to test 6, would slightly drop.
# Dive conditionNumber of nodes Time (s)60% gap Total 60% gap Total
1 In node containing the best incumbent x
⇤ -9.33% -3.27% -8.16% -0.74%
2 In node such that x
⇤b 2 [x
()b , x
(+1)
b ] -20.29% -0.35% -15.55% 1.08%3 Inverted guided dive 3.47% 1.86% 1.75% 5.65%4 With parent heuristic -13.86% -1.95% -12.45% -0.12%5 With x
⇤ or parent heuristic -25.30% -2.23% -18.55% 1.74%6 With feasible point with best obj. value -26.88% -5.81% -18.89% -2.45%7 With v
1
and v
2
-27.70% -5.31% -25.29% -8.53%
Table 1: Diving strategies
Table 2 shows how often on average the first feasible solutions (ordered byobjective value, as in test 6) are used to direct the dive at the nodes of the sBB.The current incumbent x⇤ is used if x⇤
b
2 [lb
, ub
]. If not, the 2nd best known feasiblesolution x0 is used if x0
b
2 [lb
, ub
]. If not, the 3rd best known feasible solution x00 isused if x00
b
2 [lb
, ub
] etc. The most remarkable result is the very scarce use of the2nd and next feasible solutions. The feasible solutions are spatially closed to oneanother. If x⇤
b
62 [lb
, ub
], the next feasible solutions are unlikely to have their xb
value in the range [lb
, ub
]. Still, using all the known feasible solutions results in thenoticeable improvement from test 2 to test 6.
Percentage of useCurrent incumbent 70.44%
Current 2nd best solution 1.77%Current 3rd best solution 1.17%Current 4th best solution 0.73%Current 5th best solution 0.50%Current 6th best solution 0.48%Current 7th best solution 0.27%
. . .No known feasible solution usable 23.68%
Table 2: Incumbents use on average
Guided Dive for the Spatial Branch-and-Bound 11
Table 6 details the time gain between the original Couenne and our modifiedversions test 2, test 6 and test 7, for each problem. For convenience, the best valueis depicted in bold for each problem.
Table 3 compares the time gains of the 53 hard instances. ’TL’ stands forthe Time Limit of two hours. Since the global solution f⇤ was not found withintwo hours for most problems, the 60% gap is considered to be between the firstfeasible solution f
0
found at root node and the best solution found among theoriginal Couenne and our test 2 and test 6 variants, instead of f⇤. Hence, the timeto close 60% of the gap is still related with the speed to find the first good feasiblesolutions. The three rightmost columns show the gap closed between f
0
and theaforementioned best solution known. The large number of ’0.0% TL’ occurencesfor the gap closed by Couenne means that beyond the initial feasible solution f
0
,the sBB did not manage to find any better feasible solution within the time limit.
Although this paper focuses on continuous variables, we also benchmarkedmixed-integer nonlinear problem (MINLP) instances from the MINLPlib libraryto assess the e↵ects of our algorithm on instances with both continuous and integervariables. As for the continuous problems, we let Couenne choose the variableto branch on (whether integer or continuous) and refer the reader to [3]. In theContinuous variant, we choose the diving side as in test 6 only if the branchingvariable is continuous, never modifying the diving side of integer variables. In theAll variant, we apply test 6 on both continuous and integer variables. Table 5compares the time gains of the 60 easy or medium instances, whereas Table 5compares the 21 hard instances. Since all the hard instances hit the time limit oftwo hours, the right columns of Table 5 show the gap closed instead of the totaltime gain.
By default, Couenne dives left on an integer variable if xb
� bxb
c < dxb
e � xb
and dives right otherwise. While the guided dive gives a noticeable improvementif it is performed on continuous variables, it brings no further improvement ifalso performed on integer variables. However in [12], the guided dive was quitee↵ective in MIP. While the guided dive performs well on integer variables in MIPand on continuous variables in NLP and MINLP, it has little e↵ect if performedon integer variables in MINLP. [3] benchmarked several MINLP instances withmost variable selection strategies. The comparison did not show a clear winnerand [3] stated that the performance is highly dependent on the instance. We alsorun tests with the original Couenne and several variable selection strategies (mostfractional, random, strong branching, reliability branching, three variants of the
Guided Dive for the Spatial Branch-and-Bound 13
strong branching, violation transfer) and obtained similar results. From our results,the di↵erence in gain (average on the easy and medium MINLP problems) withthe 60% primal gap and with the total running time is below 2.28% between thevariable selection strategies. We believe that the guided dive is une↵ective on theinteger variables of MINLP problems for the same reason.
60% primal gap time (s) Total time (s)Problem Coue.Cont. Gain All Gain Coue.Cont. Gain All Gainalan 0.05 0.05 4.9% 0.05 7.4% 0.08 0.07 -13.1% 0.07 -6.8%batch 0.62 0.70 10.2% 1.10 43.1% 0.63 0.70 10.2% 1.10 43.3%
60% primal gap time (s) Gap closed (%)Problem CouenneCont. Gain All Gain Couenne Cont. Allwatersbp 20.0 11.8 -41.0% 22.4 10.6% 100% TL 100% TL 100% TL
Mean -10.91% -14.25%
Table 5: Results for the MINLP hard problems
4.2 Multiway branching
Figure 5 and Figure 6 show the e↵ect of the � parameter, which corresponds tothe minimum width for a new node candidate to be accepted. The gains are anaverage over the easy and medium size problems. The 4-way partitioning is usedif all the 3 new node candidates (parts 2, 3 and 4 in Figure 4) have a minimumwidth of � (part 1 in Figure 4 is always kept as is). If not, the 3-way partitioningis considered with the same condition on the minimum width. If the condition isstill not matched by the 3-way partitioning, the 2-way partitioning is used.
If � = 0, all the new node candidates of the 4-way partitioning are alwaysconsidered wide enough. However, this does not mean that the 4-way partitioningis possible in all configurations. If h
b
is close to lb
(resp. ub
), then the margin �1
(resp. �2
) may set the branching point hb
� �1
(resp. hb
+ �2
) outside of theallowed range [l
b
, ub
], making the 4-way partitioning sometimes impossible withthe margins �
1
and �2
. If the previous situation occurs or if one node candidateis rejected because its width is below � > 0, the algorithm falls back on the 3-waypartitioning.
Unlike the 4-way partitioning, the 3-way partitioning is always possible becausethe new branching point can either be set on the left or on the right of h
b
, as longas �
1
,�2
1�↵
4
(ub
� lb
) and the nodes are not merged because of �. The tightestcondition on the maximum value of �
1
and �2
occurs if xb
has its rightmost value(i.e. the distance between x
b
and ub
is as small as possible) and hb
= ¯
xb+ub2
. Therightmost value of x
b
is obtained for xb
= ub
, which yieds xb
= 1�↵
2
lb
+ 1+↵
2
ub
. Ifhb
= ¯
xb+ub2
, the 3-way partitioning is possible if �1
,�2
ub�¯
xb2
= 1�↵
4
(ub
� lb
).Although the 3-way partitioning is possible at all nodes with this condition, the2-way partitioning must be used if one node is not wide enough because of �, if noknown incumbents have their variable value x
b
within [lb
, ub
] or if the algorithmchooses not to use any incumbent to direct the diving at a node.
Figure 5 and Figure 6 show the node and the time gain, against the nodeacceptance width �. The two plots confirm the insight that the multiway branchingshould be scarcely used. Creating too many nodes because of a low � results inmore small nodes and more computing time. As � is increased, the multiway isused less and starts reducing the number of nodes explored to achieve 60% of theprimal gap, while not significantly increasing the total number of nodes.
If part 2 is possible (Figure 4), it is considered too small if hb
��1
� xb
< �.The largest range [x
b
, hb
��1
] is obtained with xb
= lb
and hb
= ub
. The conditionbecomes 1+↵
2
(ub
� lb
) � �1
< �. If part 4 is possible (Figure 4), it is consideredtoo small if 1+↵
2
(ub
� lb
) � �2
< � with a similar reasoning. Therefore, if � >1+↵
2
� min(�1
,�2
), the parts 2 and 4 are always considered too small and thealgorithm performs a 2-way partitioning. Figure 7 plots the average repartition ofthe branching types over all the instances. With ↵ = 1
4
,�1
= 0.07(ub
� lb
),�2
=
16 D. Gerard et al.
0.13(ub
�lb
), beyond � = 1+↵
2
�min(�1
,�2
) = 0.555(ub
�lb
), the 2-way partitioning(with either the guided dive or not) is always used. Slightly below this threshold,the 3-way and 4-way partitionings are also unlikely to be used because x
b
andhb
would need to be very close to lb
and ub
. The key point is to use the 3-waybranching scarcely. Moreover, the 4-way branching is too prohibitive and shouldnot be used at all.
One parameter limits the depth at which IPOPT can be called, to preventspending too much time in the NLP solver. However, by creating 4 or 3 nodesinstead of only 2, IPOPT is called at more nodes overall. The sBB algorithm istherefore more likely to find new incumbents. As a consequence, the number ofnodes gain is improved at the cost of wasting time. This bias would prevent afair comparison between the 2-way and multiway branching procedures. We adda maximum number of times that IPOPT may be called. Depending on whetherthe 2-way, 3-way or 4-way partitionings are used at the nodes, they may lead thebranch-and-bound to di↵erent scenarios, but the overall time spent in IPOPT willbe roughly the same. This helps comparing the partitionings more fairly.
Minimum node width for acceptance (% of ub-lb)0 10 20 30 40 50 60 70 80 90 100
Nod
es g
ain
(%)
-30
-25
-20
-15
-10
-5
0
5
10
0:5 +,
2!min("1;"2)
2-way total nodes gainmultiway total nodes gain2-way 60% gap nodes gainmultiway 60% gap nodes gain
Fig. 5: Node gain for 60% primal gap and convergence with(�
1
,�2
) = (0.07, 0.13)
5 Conclusion
This paper has presented an adaptation of the guided dive for solving NLP prob-lems to global optimality in a spatial Brand-and-Bound algorithm. We first mo-tivated the original idea of using the incumbent to direct the dive even if it doesnot belong to the current node. We showed which feasible solution to use if sev-eral di↵erent feasible solutions are able to direct a dive. The heuristics describedwere benchmarked in Couenne with a wide range of easy, medium size as well ashard problems from the Coconut library. An insight of the results of the guideddive for easy, medium and hard MINLP problems from the MINLPlib library waspresented.
Guided Dive for the Spatial Branch-and-Bound 17
Minimum node width for acceptance (% of ub-lb)0 10 20 30 40 50 60 70 80 90 100
Tim
e ga
in (%
)
-25
-20
-15
-10
-5
0
5
10
0:5 +,
2!min("1;"2)
2-way total time gainmultiway total time gain2-way 60% gap time gainmultiway 60% gap time gain
Fig. 6: Time gain for 60% primal gap and convergence with(�
1
,�2
) = (0.07, 0.13)
Fig. 7: Multiway branching repartition for (�1
,�2
) = (0.07, 0.13)
An adaptative multiway branching tailored for the guided dive has been de-scribed and benchmarked. While the results of the guided dive for the 2-way par-titioning are very encouraging without the need to fine tune the parameters, theresults for the multiway branching are more mitigated. A noticeable gain can onlybe achieved if the parameters are somewhat benchmarked.
There is still room for improvement in the branching process. We focused onthe local branching around a known feasible solution, but all the known feasiblesolutions still carry more information about the whereabouts of the other feasiblesolutions. While most parameters in our heuristics need not be fine-tuned, onemay obtain better results if they could be adapted online.
18 D. Gerard et al.60%
primalga
ptime(s)
Totaltime(s)
Problem
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
alkyl
0.217
0.143
-33.87%
0.13
-40.01%
0.143
-33.87%
0.23
0.267
13.76%
0.247
6.77%
0.26
11.54%
chem
252
0.37
-99.85%
0.387
-99.85%
0.357
-99.86%
936
914
-2.32%
924
-1.28%
920
-1.68%
chen
ery
1.85
1.05
-43.52%
1.08
-41.55%
1.08
-41.90%
7.11
12.5
43.32%
12.7
43.85%
12.7
43.81%
ex1418
0.117
0.153
23.87%
0.163
28.54%
0.157
25.53%
0.12
0.157
23.42%
0.163
26.52%
0.157
23.42%
ex1428
0.41
0.44
6.82%
0.137
-66.66%
0.137
-66.66%
0.41
0.44
6.82%
0.137
-66.66%
0.137
-66.66%
ex1429
0.383
0.347
-9.55%
0.173
-54.79%
0.173
-54.79%
0.383
0.347
-9.55%
0.173
-54.79%
0.173
-54.79%
ex419
0.0533
0.0433-18.76%
0.0467
-12.38%0.0433-18.76%
0.09
0.0867
-3.67%
0.0867
-3.67%
0.0833
-7.44%
ex522ca
se1
0.173
0.153
-11.54%
0.217
20.03%
0.193
10.35%
0.223
0.203
-8.96%
0.217
-2.96%
0.193
-13.43%
ex532
0.647
0.457
-29.38%
0.463
-28.36%
0.467
-27.83%
0.647
0.457
-29.38%
0.463
-28.36%
0.467
-27.83%
ex542
0.847
0.73
-13.78%
0.717
-15.35%
0.753
-11.03%
0.847
0.73
-13.78%
0.717
-15.35%
0.753
-11.03%
ex544
0.293
0.337
12.89%
0.34
13.74%
0.347
15.40%
9.83
35
71.89%
43.8
77.58%
43.9
77.64%
ex611
18.9
17.9
-5.22%
21.5
12.45%
23.3
18.97%
106
105
-0.03%
101
-3.87%
104
-1.07%
ex612
0.07
0.0767
8.74%
0.0633
-9.57%
0.0533-23.86%
0.177
0.18
1.83%
0.17
-3.79%
0.15
-15.11%
ex613
1.95
1.25
-35.96%
1.2
-38.36%
1.18
-39.22%
410
449
8.63%
420
2.33%
457
10.12%
ex614
0.453
0.21
-53.67%
0.2
-55.88%
0.22
-51.47%
0.627
0.653
4.07%
0.593
-5.33%
0.633
1.04%
ex722
0.393
0.43
8.53%
0.403
2.48%
0.423
7.09%
0.393
0.43
8.53%
0.403
2.48%
0.423
7.09%
ex724
2.47
0.317
-87.16%
0.327
-86.76%
0.303
-87.70%
3.16
2.92
-7.59%
2.92
-7.80%
2.84
-10.22%
ex727
0.143
0.113
-20.94%
0.11
-23.24%
0.14
-2.30%
0.193
0.2
3.35%
0.19
-1.71%
0.18
-6.88%
ex728
1.82
0.363
-80.04%
0.357
-80.40%
0.34
-81.32%
3.26
3.24
-0.61%
3.22
-1.03%
3.19
-2.05%
ex729
6.94
7.53
7.88%
13.7
49.27%
13.5
48.66%
47.3
46.1
-2.60%
13.7
-71.10%
13.5
-71.44%
ex734
2.76
0.673
-75.61%
0.513
-81.40%
0.493
-82.13%
2.78
2.67
-4.07%
3.11
10.41%
37.12%
ex735
23.2
22.4
-3.36%
22.4
-3.55%
22.1
-4.91%
23.2
22.4
-3.36%
22.4
-3.55%
22.1
-4.91%
ex817
0.173
0.14
-19.22%
0.153
-11.54%
0.167
-3.81%
0.26
0.253
-2.58%
0.263
1.25%
0.257
-1.27%
ex841
11.3
0.447
-96.06%
0.447
-96.06%
0.447
-96.06%
43.4
41.5
-4.46%
41.9
-3.50%
42.4
-2.44%
ex843
8.43
8.94
5.71%
8.58
1.75%
8.96
5.99%
8.43
8.94
5.71%
8.58
1.75%
8.96
5.99%
ex844
1.13
0.4
-64.60%
0.417
-63.12%
0.413
-63.42%
2.14
2.08
-2.65%
2.18
2.13%
2.19
2.43%
ex845
0.24
0.243
1.36%
0.22
-8.33%
0.207
-13.87%
6.81
7.06
3.54%
5.57
-18.30%
5.51
-19.18%
ex853
0.61
0.207
-66.11%
0.213
-65.03%
0.197
-67.75%
1.2
1.32
9.07%
1.36
11.74%
1.19
-0.83%
ex854
0.28
0.167
-40.46%
0.163
-41.68%
0.153
-45.25%
2.18
2.3
5.22%
2.26
3.69%
2.41
9.80%
ex855
0.177
0.0867-50.93%
0.0867-50.93%
0.0933
-47.20%
1.14
1.39
17.75%
1.1
-3.79%
1.21
5.77%
ex924
0.03
0.03
0.00%
0.0267-11.00%
0.03
0.00%
0.03
0.03
0.00%
0.0267-11.00%
0.03
0.00%
expfit
0.35
0.38
7.89%
0.353
0.93%
0.37
5.41%
0.35
0.38
7.89%
0.353
0.93%
0.37
5.41%
expfitb
152
162
6.15%
166
8.59%
165
7.83%
152
162
6.15%
166
8.59%
165
7.83%
fccu
0.19
0.197
3.41%
0.203
6.54%
0.2
5.00%
0.19
0.197
3.41%
0.203
6.54%
0.2
5.00%
fletch
er0.503
0.123
-75.50%
0.137
-72.84%
0.12
-76.16%
1.03
0.813
-21.29%
0.83
-19.67%
0.95
-8.06%
Guided Dive for the Spatial Branch-and-Bound 1960%
primalga
ptime(s)
Totaltime(s)
Problem
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
gtm
0.33
0.327
-1.00%
0.31
-6.06%
0.313
-5.06%
0.33
0.327
-1.00%
0.31
-6.06%
0.313
-5.06%
haifas
11.4
11.1
-2.52%
10.7
-6.27%
10.6
-6.80%
34.3
34.3
-0.17%
28.5
-16.97%
46.2
25.73%
haldmad
s11.6
1.5
-87.11%
1.41
-87.88%
1.22
-89.54%
11.6
1.5
-87.11%
1.41
-87.88%
1.22
-89.54%
han
ging
107
106
-1.63%
105
-2.56%
106
-1.31%
107
106
-1.63%
105
-2.56%
106
-1.31%
harker
1.45
1.09
-25.06%
1.09
-24.60%
1.03
-28.97%
1.45
1.09
-25.06%
1.09
-24.60%
1.03
-28.97%
hart6
1.17
1.08
-7.71%
1.08
-7.43%
1.11
-5.14%
1.17
1.08
-7.71%
1.08
-7.43%
1.11
-5.14%
hatfldh
0.04
0.0367
-8.25%
0.03
-25.00%
0.0367
-8.25%
0.04
0.0367
-8.25%
0.03
-25.00%
0.0367
-8.25%
hav
erly
0.28
0.277
-1.18%
0.257
-8.32%
0.26
-7.14%
0.28
0.277
-1.18%
0.257
-8.32%
0.26
-7.14%
hhfair
7.85
1.12
-85.70%
1.05
-86.67%
1.07
-86.38%
12.8
22.3
42.50%
21.1
39.45%
21.4
40.22%
him
mel16
18.1
0.527
-97.09%
0.533
-97.05%
0.54
-97.01%
28.8
13.1
-54.59%
12
-58.21%
12
-58.44%
him
melbk
1.96
2.2
10.93%
2.16
9.27%
2.13
7.99%
1.96
2.2
10.93%
2.16
9.27%
2.13
7.99%
him
melp1
0.927
0.123
-86.69%
0.13
-85.97%
0.12
-87.05%
1.36
1.39
2.16%
1.39
1.93%
1.36
0.24%
him
melp2
0.933
0.987
5.41%
0.14
-85.00%
0.137
-85.35%
1.53
1.56
2.34%
1.49
-2.62%
1.49
-2.62%
him
melp3
0.313
0.127
-59.56%
0.117
-62.75%
0.12
-61.70%
0.313
0.34
7.85%
0.34
7.85%
0.337
6.95%
him
melp4
0.41
0.16
-60.98%
0.153
-62.61%
0.16
-60.98%
0.417
0.423
1.56%
0.433
3.83%
0.433
3.83%
him
melp5
0.397
0.36
-9.25%
0.34
-14.29%
0.353
-10.94%
0.397
0.36
-9.25%
0.34
-14.29%
0.353
-10.94%
him
melp6
0.477
0.163
-65.74%
0.15
-68.53%
0.147
-69.23%
0.477
0.517
7.74%
0.513
7.13%
0.5
4.66%
hou
se0.25
0.23
-8.00%
0.377
33.63%
1.04
75.96%
2.71
3.12
13.25%
2.67
-1.48%
2.85
5.03%
hs005
0.03
0.0267
-11.00%0.0233-22.33%
0.0233-22.33%
0.03
0.0267
-11.00%0.0233-22.33%
0.0233-22.33%
hs010
0.663
0.493
-25.63%
0.49
-26.13%
0.5
-24.62%
0.663
0.493
-25.63%
0.49
-26.13%
0.5
-24.62%
hs012
0.0233
0.0267
12.73%
0.0233
0.00%
0.0167-28.33%
0.0233
0.0333
30.03%
0.0267
12.73%
0.02
-14.16%
hs029
0.0633
0.237
73.26%
0.0467-26.22%
0.0467-26.22%
0.25
0.253
1.30%
0.247
-1.32%
0.24
-4.00%
hs035
0.0667
0.0467
-29.99%0.0433-35.08%
0.0433-35.08%
0.177
0.183
3.60%
0.18
1.83%
0.167
-5.66%
hs037
0.117
0.103
-11.48%
0.0833
-28.62%
0.08
-31.45%
0.117
0.103
-11.48%
0.0833
-28.62%
0.08
-31.45%
hs038
0.23
0.103
-55.09%
0.113
-50.74%
0.11
-52.17%
0.463
0.5
7.34%
0.497
6.72%
0.44
-5.03%
hs040
0.0567
0.08
29.12%
0.0767
26.08%
0.07
19.00%
0.0567
0.08
29.12%
0.0767
26.08%
0.07
19.00%
hs041
0.06
0.0433-27.83%
0.0667
10.04%
0.0533
-11.17%
0.06
0.0433-27.83%
0.0667
10.04%
0.0533
-11.17%
hs054
0.0433
0.0333
-23.09%
0.03
-30.72%
0.0333
-23.09%
0.0433
0.0333
-23.09%
0.03
-30.72%
0.0333
-23.09%
hs057
6.85
2.42
-64.64%
2.39
-65.17%
2.39
-65.17%
6.85
2.43
-64.59%
2.39
-65.13%
2.39
-65.17%
hs060
0.0333
0.04
16.75%
0.0333
0.00%
0.0367
9.26%
0.0333
0.04
16.75%
0.0333
0.00%
0.0367
9.26%
hs062
0.477
0.0833
-82.53%0.0733-84.62%
0.1
-79.02%
0.973
1.01
3.95%
1.09
10.44%
1.15
15.37%
hs063
0.05
0.05
0.00%
0.0433-13.40%
0.0467
-6.60%
0.05
0.05
0.00%
0.0433-13.40%
0.0467
-6.60%
hs071
0.197
0.17
-13.57%
0.19
-3.41%
0.187
-5.08%
0.197
0.17
-13.57%
0.19
-3.41%
0.187
-5.08%
hs073
0.04
0.04
0.00%
0.04
0.00%
0.04
0.00%
0.04
0.04
0.00%
0.04
0.00%
0.04
0.00%
hs074
3.68
3.44
-6.52%
1.19
-67.69%
1.23
-66.52%
18.7
17.9
-4.05%
8.74
-53.20%
8.43
-54.88%
20 D. Gerard et al.60%
primalga
ptime(s)
Totaltime(s)
Problem
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
hs075
0.1
0.0867
-13.30%0.0833-16.70%
0.09
-10.00%
0.1
0.0867
-13.30%0.0833-16.70%
0.09
-10.00%
hs077
0.18
0.18
0.00%
0.183
1.80%
0.167
-7.39%
0.18
0.18
0.00%
0.183
1.80%
0.167
-7.39%
hs078
0.3
0.177
-41.10%
0.18
-40.00%
0.163
-45.57%
1.57
1.69
7.10%
1.63
3.68%
1.62
3.09%
hs079
0.123
0.113
-8.11%
0.11
-10.79%
0.1
-18.90%
0.123
0.113
-8.11%
0.11
-10.79%
0.1
-18.90%
hs080
1.47
0.35
-76.24%
0.923
-37.33%
0.96
-34.84%
1.47
0.35
-76.24%
0.923
-37.33%
0.96
-34.84%
hs081
1.02
1-1.64%
0.923
-9.19%
0.857
-15.74%
1.02
1-1.64%
0.923
-9.19%
0.857
-15.74%
hs086
0.38
0.2
-47.37%
0.21
-44.74%
0.197
-48.24%
0.67
0.6
-10.45%
0.607
-9.45%
0.553
-17.42%
hs099
5.59
4.45
-20.29%
638
99.12%
633
99.12%
646
640
-0.83%
638
-1.22%
633
-1.92%
hs100
0.447
0.533
16.24%
0.523
14.64%
0.53
15.72%
0.503
0.533
5.63%
0.523
3.82%
0.53
5.04%
hs100mod
0.31
0.35
11.43%
0.31
0.00%
0.183
-40.87%
0.347
0.35
0.94%
0.31
-10.59%
0.323
-6.75%
hs104
30.34
-88.65%
0.343
-88.54%
0.333
-88.88%
4.14
3.63
-12.17%
3.6
-13.05%
3.66
-11.52%
hs106
1.39
1.22
-11.99%
2.89
51.90%
2.85
51.17%
2.97
3.01
1.33%
2.89
-2.69%
2.85
-4.15%
hs108
2.66
22.6
88.26%
23.2
88.55%
0.417
-84.32%
671
771
12.98%
724
7.38%
739
9.27%
hs113
0.577
0.587
1.70%
0.49
-15.03%
0.51
-11.57%
0.577
0.587
1.70%
0.49
-15.03%
0.51
-11.57%
hs116
0.383
0.387
0.88%
0.377
-1.72%
0.36
-6.08%
0.6
0.583
-2.78%
0.587
-2.22%
0.57
-5.00%
hs119
6.5
0.8
-87.70%
0.817
-87.44%
0.81
-87.54%
8.15
8.71
6.43%
8.69
6.14%
8.68
6.03%
hs35mod
0.05670.0567
0.00%
0.0733
22.65%
0.0633
10.43%
0.07
0.07
0.00%
0.0733
4.50%
0.0633
-9.57%
launch
1.34
1.35
0.74%
1.3
-2.74%
1.28
-4.23%
1.97
1.97
-0.17%
2.29
13.83%
2.24
12.04%
mea
nva
r0.92
0.153
-83.34%
0.15
-83.70%
0.147
-84.05%
0.937
0.903
-3.57%
0.893
-4.63%
0.873
-6.77%
mhw4d
0.13
0.117
-10.23%
0.127
-2.54%
0.113
-12.85%
0.203
0.223
8.96%
0.24
15.29%
0.21
3.19%
minlphi
0.833
0.82
-1.60%
0.797
-4.39%
0.81
-2.80%
1.06
1.08
1.55%
1.06
-0.31%
1.07
0.63%
mwright
0.0633
0.05
-21.01%
0.0367-42.02%
0.0467
-26.22%
0.197
0.147
-25.42%
0.14
-28.83%
0.153
-22.06%
obstclal
7.61
7.18
-5.65%
7.44
-2.32%
7.56
-0.70%
7.61
7.18
-5.65%
7.44
-2.32%
7.56
-0.70%
palmer2
0.373
39.6
99.06%
60.8
99.39%
31.2
98.80%
0.377
39.6
99.05%
60.8
99.38%
31.2
98.79%
palmer3c
0.06
0.06
0.00%
0.06
0.00%
0.06
0.00%
0.06
0.06
0.00%
0.06
0.00%
0.0633
5.21%
palmer5d
0.05
0.0567
11.82%
0.06
16.67%
0.06
16.67%
0.05
0.0567
11.82%
0.06
16.67%
0.06
16.67%
palmer8c
0.0867
0.09
3.67%
0.0667-23.07%
0.0667-23.07%
0.0867
0.09
3.67%
0.0667-23.07%
0.0667-23.07%
polak3
2.59
2.58
-0.39%
2.6
0.26%
2.71
4.42%
2.59
2.58
-0.39%
2.6
0.26%
2.71
4.42%
process
37.1
26.9
-27.37%
0.13
-99.65%
0.127
-99.66%
113
5346
-69.26%
53.9
-95.21%
54.6
-95.15%
prodpl0
0.21
0.203
-3.19%
0.207
-1.57%
0.217
3.09%
0.21
0.203
-3.19%
0.207
-1.57%
0.217
3.09%
prodpl1
0.37
0.307
-17.11%
0.313
-15.32%
0.307
-17.11%
0.463
0.453
-2.16%
0.47
1.43%
0.463
0.00%
qcq
p2447
80.0233
0.02
-14.16%
0.0167-28.33%
0.0233
0.00%
0.0233
0.02
-14.16%
0.0167-28.33%
0.0233
0.00%
qcq
p2721
20.06
0.07
14.29%
0.06
0.00%
0.0567
-5.50%
0.06
0.07
14.29%
0.06
0.00%
0.0567
-5.50%
qcq
p2744
20.09
0.0967
6.93%
0.0967
6.93%
0.0933
3.54%
0.0933
0.0967
3.52%
0.0967
3.52%
0.0933
0.00%
qpcb
oei2
6.33
6.35
0.37%
6.08
-3.90%
6.04
-4.53%
7.25
7.29
0.50%
6.08
-16.09%
6.04
-16.64%
Guided Dive for the Spatial Branch-and-Bound 2160%
primalga
ptime(s)
Totaltime(s)
Problem
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
Cou
enneTest2
Gain
Test6
Gain
Test7
Gain
qpnboei2
11.2
9.72
-13.50%
12.8
12.13%
12.8
11.94%
20.4
17.4
-14.89%
12.8
-37.34%
12.8
-37.47%
rk23
1.11
3.16
64.81%
3.16
64.81%
3.15
64.66%
1.12
3.18
64.85%
3.17
64.81%
3.16
64.66%
robot
113
108
-4.31%
0.413
-99.63%
0.403
-99.64%
212
214
0.70%
170
-20.00%
192
-9.70%
sambal
0.09330.0933
0.00%
0.0933
0.00%
0.0967
3.52%
0.09330.0933
0.00%
0.0933
0.00%
0.0967
3.52%
sisser
0.383
0.373
-2.61%
0.367
-4.33%
0.347
-9.55%
0.383
0.373
-2.61%
0.367
-4.33%
0.347
-9.55%
span
hyd
4.5
4.52
0.52%
5.48
17.99%
5.49
18.04%
4.5
4.52
0.52%
5.48
17.99%
5.49
18.04%
tointqor
1.66
1.7
2.54%
1.62
-2.41%
1.69
1.58%
1.66
1.7
2.54%
1.62
-2.41%
1.69
1.58%
trim
loss
2.94
2.98
1.23%
2.99
1.67%
2.96
0.56%
2.94
2.98
1.23%
2.99
1.67%
2.96
0.56%
twobars
0.04
0.04
0.00%
0.0367
-8.25%
0.0333-16.75%
0.0467
0.0433
-7.28%
0.04
-14.35%
0.0367-21.41%
wall
3.98
3.86
-2.93%
3.87
-2.85%
3.97
-0.33%
3.98
3.86
-2.93%
3.87
-2.85%
3.97
-0.33%
zecevic4
0.03
0.03
0.00%
0.0233-22.33%
0.0333
9.91%
0.03
0.03
0.00%
0.0233-22.33%
0.0333
9.91%
zy2
0.06
0.0733
18.14%
0.0633
5.21%
0.06
0.00%
0.06
0.0733
18.14%
0.0633
5.21%
0.06
0.00%
Tab
le6:
Resultsfortheeasy
andmed
ium
size
problems
22 D. Gerard et al.
Appendix
See Table 6.
References
1. P. Abichandani, H. Y. Benson, and M. Kam. Multi-vehicle path coordination under com-munication constraints. In Proceedings of the American Control Conference, pages 650–656, 2008.
2. C. Adjiman, S. Dallwig, C. Floudas, and A. Neumaier. A global optimization method,alpha bb, for general twice-di↵erentiable constrained NLPs - I. theoretical advances. Com-puters & Chemical Engineering, 22:1137–1158, 1998.
3. P. Belotti, J. Lee, L. Liberti, F. Margot, and A. Wachter. Branching and bounds tighteningtechniques for non-convex MINLP. Optimization Methods and Software, 24(4-5):597–634,2009.
5. T. Berthold, G. Gamrath, A. M. Gleixner, S. Heinz, T. Koch, and Y. Shinano. Solvingmixed integer linear and nonlinear problems using the SCIP optimization suite. TechnicalReport 12-27, ZIB, 2012.
6. L. Biegler. Nonlinear Programming: Concepts, Algorithms, and Applications to ChemicalProcesses. Society for Industrial and Applied Mathematics, 2010.
7. L. T. Biegler. System Modeling and Optimization: 23rd IFIP TC 7 Conference, Cracow,Poland, July 23-27, 2007, Revised Selected Papers, chapter E�cient Nonlinear Program-ming Algorithms for Chemical Process Control and Operations, pages 21–35. SpringerBerlin Heidelberg, 2009.
8. P. T. Boggs and J. W. Tolle. Sequential quadratic programming. Acta Numerica, 4:1–51,1 1995.
9. R. H. Byrd, R. B. Schnabel, and G. A. Shultz. A trust region algorithm for nonlinearlyconstrained optimization. SIAM Journal on Numerical Analysis, 24(5):1152–1170, 1987.
10. T. T. Chung and T. C. Sun. Weight optimization for flexural reinforced concrete beamswith static nonlinear response. Structural optimization, 8(2):174–180, 1994.
11. E. Costa-Montenegro, F. J. Gonzalez-Castano, P. S. Rodrıguez-Hernandez, and J. C.Burguillo-Rial. Computational Science – ICCS 2007: 7th International Conference, Bei-jing, China, May 27 - 30, 2007, Proceedings, Part IV, chapter Nonlinear Optimization ofIEEE 802.11 Mesh Networks, pages 466–473. Springer Berlin Heidelberg, 2007.
12. E. Danna, E. Rothberg, and C. L. Pape. Exploring relaxation induced neighborhoods toimprove MIP solutions. Mathematical Programming, 2005.
13. V. Donde, V. Lopez, B. Lesieutre, A. Pinar, C. Yang, and J. Meza. Identification of severemultiple contingencies in electric power networks. In Power Symposium, 2005. Proceedingsof the 37th Annual North American, pages 59–66, 2005.
14. A. V. Fiacco and G. P. McCormick. Nonlinear programming; sequential unconstrainedminimization techniques. Wiley New York, 1968.
15. C. Fielding. Advanced techniques for clearance of flight control laws, volume 283 of Engi-neering online library. Springer, 2002.
16. M. Fischetti and A. Lodi. Local branching. Mathematical Programming, 98(1):23–47,2003.
17. R. Fletcher and S. Ley↵er. Nonlinear programming without a penalty function. Mathe-matical Programming, 91(2):239–269, 2002.
18. A. Fugenschuh, M. Herty, A. Klar, and A. Martin. Combinatorial and continuous mod-els for the optimization of tra�c flows on networks. SIAM Journal on Optimization,16(4):1155–1176, 2006.
19. E. P. Gatzke, J. E. Tolsma, and P. I. Barton. Construction of convex relaxations usingautomated code generation techniques. Optimization and Engineering, 3(3):305–326, 2002.
20. B. H. Gebreslassie, M. Slivinsky, B. Wang, and F. You. Life cycle optimization for sus-tainable design and operations of hydrocarbon biorefinery via fast pyrolysis, hydrotreatingand hydrocracking. Computers & Chemical Engineering, 50:71 – 91, 2013.
21. I. Gentilini, F. Margot, and K. Shimada. The travelling salesman problem with neigh-bourhoods: MINLP solution. Optimization Methods and Software, 28(2):364–378, 2013.
Guided Dive for the Spatial Branch-and-Bound 23
22. M. Isshiki, D. Sinclair, and S. Kaneko. Lens design: Global optimization of both per-formance and tolerance sensitivity. In International Optical Design. Optical Society ofAmerica, 2006.
23. J. R. Jackson, J. Hofmann, J. Wassick, and I. E. Grossmann. A nonlinear multiperiodprocess optimization model for production planning in multi-plant facilities. In ProceedingsFOCAPO 2003, pages 281–284, 2003.
24. R. Karuppiah and I. Grossmann. Global optimization for the synthesis of integrated watersystems in chemical processes. Computers & Chemical Engineering, 30:650–673, 2006.
25. L. Liberti, G. Nannicini, and N. Mladenovic. A Good Recipe for Solving MINLPs, pages231–244. Springer US, Boston, MA, 2010.
26. Y. Lin and L. Schrage. The global solver in the LINDO API. Optimization MethodsSoftware, 24(4-5):657–668, 2009.
27. R. Misener and C. A. Floudas. Antigone: Algorithms for continuous / integer globaloptimization of nonlinear equations. J. of Global Optimization, 59(2-3):503–526, July2014.
28. N. Mladenovic and P. Hanse. Variable neighborhood search. Computers and OperationsResearch.
29. J. Momoh, R. Koessler, M. Bond, B. Stott, D. Sun, A. Papalexopoulos, and P. Ristanovic.Challenges to optimal power flow. Power Systems, IEEE Transactions on, 12(1):444–455,1997.
30. G. Nannicini and P. Belotti. Rounding-based heuristics for nonconvex minlps. Mathemat-ical Programming Computation, 4(1):1–31, 2012.
31. G. Nannicini, P. Belotti, and L. Liberti. A local branching heuristic for MINLPs. ArXive-prints, 2008.
32. Y. Nesterov and A. Nemirovski. Interior-point polynomial algorithms in convex program-ming. SIAM studies in applied mathematics. Society for Industrial and Applied Mathe-matics, 1994.
33. A. Neumaier. Complete search in continuous global optimization and constraint satisfac-tion, acta numerica 13, 2004.
34. J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 2nd edition, 2006.35. M. E. Pfetsch, A. Fugenschuh, B. Geisler, N. Geisler, R. Gollmer, B. Hiller, J. Humpola,
T. Koch, T. Lehmann, A. Martin, A. Morsi, J. Rovekamp, L. Schewe, M. Schmidt,R. Schultz, R. Schwarz, J. Schweiger, C. Stangl, M. C. Steinbach, S. Vigerske, and B. M.Willert. Validation of nominations in gas network optimization: Models, methods, andsolutions. Optimization Methods and Software, 2014.
36. A. Quist, R. van Geemert, J. Hoogenboom, T. Illes, C. Roos, and T. Terlaky. Application ofnonlinear optimization to reactor core fuel reloading. Annals of Nuclear Energy, 26(5):423– 448, 1999.
37. A. U. Raghunathan, V. Gopal, D. Subramanian, L. T. Biegler, and T. Samad. Dynamic op-timization strategies for three-dimensional conflict resolution of multiple aircraft. Journalof guidance, control, and dynamics, 27(4):586–594, 2004.
38. E. Smith and C. Pantelides. A symbolic reformulation/spatial branch-and-bound algo-rithm for the global optimisation of nonconvex MINLPs. Computers & Chemical Engi-neering, 23(4-5):457 – 478, 1999.
39. M. Soleimanipour, W. Zhuang, and G. Freeman. Optimal resource management in wire-less multimedia wideband CDMA systems. Mobile Computing, IEEE Transactions on,1(2):143–160, 2002.
40. M. Tawarmalani and N. Sahinidis. Convexification and Global Optimization in Con-tinuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, andApplications. Nonconvex Optimization and Its Applications. Springer US, 2002.
41. M. Tawarmalani and N. V. Sahinidis. A polyhedral branch-and-cut approach to globaloptimization. Mathematical Programming, 103:225–249, 2005.
